Integrable structure of modified melting crystal model

Our previous work on a hidden integrable structure of the melting crystal model (the U(1) Nekrasov function) is extended to a modified crystal model. As in the previous case, “shift symmetries” of a quantum torus algebra plays a central role. With the aid of these algebraic relations, the partition function of the modified model is shown to be a tau function of the 2D Toda hierarchy. We conjecture that this tau function belongs to a class of solutions (the so called Toeplitz reduction) related to the Ablowitz-Ladik hierarchy.

💡 Research Summary

The paper extends the hidden integrable structure discovered in the original melting crystal model—equivalently the U(1) Nekrasov partition function—to a modified crystal model that incorporates asymmetric weight factors. The authors begin by recalling that the original model’s partition function can be expressed as a vacuum expectation value of exponentiated free‑fermion currents dressed by an element of the quantum torus algebra. The crucial algebraic tool is the “shift symmetry” of the quantum torus algebra, which governs the commutation relations of the generators (V_{m,n}) and allows one to move exponential factors freely across the vacuum state.

In the modified model each box in a three‑dimensional Young diagram carries an extra weight (q^{\alpha i+\beta j}) depending on its horizontal coordinates. This deformation translates into a modified set of torus generators (\widetilde{V}{m,n}=q^{\alpha m}V{m,n}). The authors prove that despite the deformation the shift symmetry remains intact: the conjugation by the basic torus operators still produces simple multiplicative factors, and the algebra closes in the same way as in the undeformed case.

Using these algebraic relations the partition function (Z_{\text{mod}}(t,\bar t)) of the modified model is rewritten as
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